Program

Let \(Z\) be an isotropic stable process in the Euclidean space. The process \(Z\) is killed upon exiting an open set \(D\) and the killed process is then subordinated by an independent \(\gamma\)-stable subordinator, \(0<\gamma <1\). The resulting process is a Hunt process in \(D\). In this talk, I will discuss several potential theoretical properties of this process such as Harnack inequality for nonnegative harmonic functions, the Carleson estimate, Green function and jumping kernel estimates in smooth sets \(D\), and in particular, the boundary Harnack principle. Surprisingly, it turns out the BHP holds only if \(1/2<\gamma<1\). This is joint work with Panki Kim and Renming Song.

Fractional and anomalous diffusions have a long history. The terms fractional and anomalous have been considered with different meaning and in different contexts. By fractional diffusion we mean a diffusion in a medium with fractional dimension (fractals, for instance) whereas, by anomalous diffusions, according to the most significant literature, we refer to a motion whose mean squared displacement is proportional to a power of time. In this context we usually have the characterization given in terms of subdiffusion/superdiffusion or normal diffusion. Our aim is to pay exclusive attention to the probabilistic models for anomalous dynamics (not necessarily anomalous diffusions). The involved processes are guided by fractional (in time/space) equations and are obtained through random time changes.

We introduce a precise definition of delayed and rushed processes and provide some examples which are, in some cases, counter-intuitive.

We consider time changes given by subordinators and their inverse processes. Our analysis shows that, quite surprisingly, inverse processes are not necessarily leading to delayed processes.

We prove that a space inversion property of a standard Markov process X implies the existence of a Kelvin transform of X-harmonic, excessive and operator–harmonic functions and that the inversion property is inherited by Doob h-transforms. We determine new classes of processes having space inversion properties amongst those satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. Some examples are treated in details.

I will consider a class of non-local equations, assuming that their solutions have a global maximum, and discuss estimates on their position. Such points are of special interest as they describe where hot-spots settle on the long run or, in a probabilistic description, the region around which paths typically concentrate. I will address, on the one hand, the interplay of competing parameters whose compromise the resulting location is, and the impact of geometric constraints, on the other.

References
[1] A. Biswas and J. Lőrinczi. Universal constraints on the location of extrema of eigenfunctions of non-local Schrödinger operators, arXiv:1710.11596, 2017 (under review)
[2] A. Biswas and J. Lőrinczi. Maximum principles and Aleksandrov-Bakelman-Pucci type estimates for non-local Schrödinger equations with exterior conditions, arXiv:1711.09267, 2017 (under review)
[3] A. Biswas and J. Lőrinczi. Maximum principles for time-fractional Cauchy problems with spatially non-local components, arXiv:1801.02349, 2018 (under review)
[4] A. Biswas and J. Lőrinczi. Ambrosetti-Prodi type results for Dirichlet problems of the fractional Laplacian, arXiv:1803.08540, 2018 (under review)

In this talk, I will discuss ergodic properties of diffusion processes focusing on the rate of convergence of the marginals of the process to the invariant measure with respect to the total variation distance and Wasserstein distance. In particular, I will present sharp conditions in terms of the coefficients of the process (generator) ensuring subexponential rate of convergence. I will also discuss ergodic properties of a class of jump processes obtained through subordination of diffusion processes.

Consider the Caputo evolution equation (EE) \(\partial^\beta_t u = \Delta u\) with initial condition \(\phi\) on \(\{0\}\times\mathbb R^d\). As it is well known, the solution reads \(u(t, x) = \mathbf{E}_x\left[\phi (B_{E_t})\right]\). Here \(B_t\) is a Brownian motion and the independent time change \(E_t\) is an inverse \(\beta\)-stable subordinator. This is a popular model for subdiffusion [6], with remarkable universality properties [1]. We substitute the Caputo derivative \(\partial^\beta_t\) with the Marchaud derivative. This results in a natural extension of the Caputo EE featuring a time-nonlocal initial condition \(\phi\) on \((-\infty, 0] \times \mathbb{R}^d\). We derive the new stochastic representation for the solution, namely \(u(t, x) = \mathbf{E}_x\left[\phi(-W_t, B_{E_t})\right]\). This stochastic representation has a pleasing interpretation due to the non-obvious presence of \(W_t\). Here \(W_t\) denotes the waiting/trapping time of the subdiffusion \(B_{E_t}\). We discuss classical-wellposedness for the space-fractional case, following [7]. Additionally, following [3, 4], we discuss weak-wellposedness for the respective extensions of Caputo-type EEs (such as in [2, 5]).

References
[1] Barlow, Martin T.; Cerny, Jiri (2011). Probability theory and related fields, 149.3-4: 639-673.
[2] Chen Z-Q., Kim P., Kumagai T., Wang J. (2017). arXiv:1708.05863.
[3] Du Q., Yang V., Zhou Z. (2017). Discrete and
continuous dynamical systems series B, Vol 22, n. 2.
[4] Du Q., Toniazzi L., Zhou Z. (2018).
Preprint. Expected submission date: Sept. 2018.
[5] Hernández-Hernández, M.E., Kolokoltsov, V.N., Toniazzi, L. (2017). Chaos, Solitons & Fractals, 102, 184-196.
[6] Meerschaert, M.M., Sikorskii, A. (2012). Stochastic Models for Fractional Calculus, De Gruyter
Studies in Mathematics, Book 43.
[7] Toniazzi L. (2018). To appear in J Math Anal Appl. arXiv:1805.02464.

In 1967 Aronson showed that the fundamental solution of the heat equation for a second order uniformly elliptic operator in divergence form satisfies two-sided Gaussian estimates. Using this celebrated result, we investigate two-sided estimates for the fundamental solution of the fractional analogues of the heat equation, where one replaces the time derivative with a Caputo fractional derivative of order \(\beta \in (0, 1)\) and also replace the second order elliptic operator with a homogeneous pseudo-differential operator. The starting point for these estimates is given by a formula, which is due to Zolotarev, that links Mittag-Leffler functions with \(\beta\)-stable densities via the Laplace transform. Probabilistically speaking, the solution of such fractional evolution equations is typically some time-changed Brownian motion, or time-changed stable process. This is joint work with Vassili Kolokoltsov.

The Generalized Elastic Model accounts for the dynamics of several physical systems, such as polymers, fluctuating interfaces, growing surfaces, membranes, proteins and file systems among others. We derive the fractional stochastic equation governing the motion of a probe particle (tracer) in such kind of systems. This Langevin equation involves the use of fractional derivative in time and satisfies the Fluctuation-Dissipation relation, it goes under the name of Fractional Langevin Equation. Within this framework the spatial correlations appearing in the Generalized Elastic Model are translated into time correlations described by the fractional derivative together with the space-time correlations of the fractional Gaussian noise. We derive the exact scaling analytical form of several physical observables such as structure factors, roughness and mean square displacement. Special attention will be devoted to the dependence on initial conditions and linear-response relations in the case of an applied potential.